Question: Suppose we have a vector-valued function $g(t)$ and a scalar function $f(x, y, z)$. Let $h(t) = f(g(t))$. We know: $\begin{aligned} &g(2) = (4, 1, 0) \\ \\ &g'(2) = (3, 5, -1) \\ \\ &\nabla f(4, 1, 0) = (0, 4, 2) \end{aligned}$ Evaluate $\dfrac{d h}{d t}$ at $t = 2$. $h'(2)=$
Solution: Formula The multivariable chain rule says that $\dfrac{dh}{dt} = \nabla f(g(t)) \cdot g'(t)$. The $g'(t)$ part is how much a change in $t$ will cause the input to $f$ to move, and the $\nabla f(g(t))$ part is how much $f$ will change in response to this update to its input. [What's the intuition behind the formula?] Applying the formula We want to find $h'(2) = \nabla f(g(2)) \cdot g'(2)$. We know the following. $\begin{aligned} &g(2) = (4, 1, 0) \\ \\ &g'(2) = (3, 5, -1) \\ \\ &\nabla f(4, 1, 0) = (0, 4, 2) \end{aligned}$ Substituting: $h'(2) = (0, 4, 2) \cdot (3, 5, -1) = 18$ Answer Therefore, $h'(2) = 18$.